diff -r 2a07222e2a8b -r 6bb15b8e6301 AFP-Submission/Lexer.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/AFP-Submission/Lexer.thy	Tue May 24 11:36:21 2016 +0100
@@ -0,0 +1,493 @@
+(*  Title:       POSIX Lexing with Derivatives of Regular Expressions
+    Authors:     Fahad Ausaf <fahad.ausaf at icloud.com>, 2016
+                 Roy Dyckhoff <roy.dyckhoff at st-andrews.ac.uk>, 2016
+                 Christian Urban <christian.urban at kcl.ac.uk>, 2016
+    Maintainer:  Christian Urban <christian.urban at kcl.ac.uk>
+*) 
+
+theory Lexer
+  imports Derivatives
+begin
+
+section {* Values *}
+
+datatype 'a val = 
+  Void
+| Atm 'a
+| Seq "'a val" "'a val"
+| Right "'a val"
+| Left "'a val"
+| Stars "('a val) list"
+
+
+section {* The string behind a value *}
+
+fun 
+  flat :: "'a val \<Rightarrow> 'a list"
+where
+  "flat (Void) = []"
+| "flat (Atm c) = [c]"
+| "flat (Left v) = flat v"
+| "flat (Right v) = flat v"
+| "flat (Seq v1 v2) = (flat v1) @ (flat v2)"
+| "flat (Stars []) = []"
+| "flat (Stars (v#vs)) = (flat v) @ (flat (Stars vs))" 
+
+lemma flat_Stars [simp]:
+ "flat (Stars vs) = concat (map flat vs)"
+by (induct vs) (auto)
+
+section {* Relation between values and regular expressions *}
+
+inductive 
+  Prf :: "'a val \<Rightarrow> 'a rexp \<Rightarrow> bool" ("\<turnstile> _ : _" [100, 100] 100)
+where
+ "\<lbrakk>\<turnstile> v1 : r1; \<turnstile> v2 : r2\<rbrakk> \<Longrightarrow> \<turnstile> Seq v1 v2 : Times r1 r2"
+| "\<turnstile> v1 : r1 \<Longrightarrow> \<turnstile> Left v1 : Plus r1 r2"
+| "\<turnstile> v2 : r2 \<Longrightarrow> \<turnstile> Right v2 : Plus r1 r2"
+| "\<turnstile> Void : One"
+| "\<turnstile> Atm c : Atom c"
+| "\<turnstile> Stars [] : Star r"
+| "\<lbrakk>\<turnstile> v : r; \<turnstile> Stars vs : Star r\<rbrakk> \<Longrightarrow> \<turnstile> Stars (v # vs) : Star r"
+
+inductive_cases Prf_elims:
+  "\<turnstile> v : Zero"
+  "\<turnstile> v : Times r1 r2"
+  "\<turnstile> v : Plus r1 r2"
+  "\<turnstile> v : One"
+  "\<turnstile> v : Atom c"
+(*  "\<turnstile> vs : Star r"*)
+
+lemma Prf_flat_lang:
+  assumes "\<turnstile> v : r" shows "flat v \<in> lang r"
+using assms
+by(induct v r rule: Prf.induct) (auto)
+
+lemma Prf_Stars:
+  assumes "\<forall>v \<in> set vs. \<turnstile> v : r"
+  shows "\<turnstile> Stars vs : Star r"
+using assms
+by(induct vs) (auto intro: Prf.intros)
+
+lemma Star_string:
+  assumes "s \<in> star A"
+  shows "\<exists>ss. concat ss = s \<and> (\<forall>s \<in> set ss. s \<in> A)"
+using assms
+by (metis in_star_iff_concat set_mp)
+
+lemma Star_val:
+  assumes "\<forall>s\<in>set ss. \<exists>v. s = flat v \<and> \<turnstile> v : r"
+  shows "\<exists>vs. concat (map flat vs) = concat ss \<and> (\<forall>v\<in>set vs. \<turnstile> v : r)"
+using assms
+apply(induct ss)
+apply(auto)
+apply (metis empty_iff list.set(1))
+by (metis concat.simps(2) list.simps(9) set_ConsD)
+
+lemma L_flat_Prf1:
+  assumes "\<turnstile> v : r" shows "flat v \<in> lang r"
+using assms
+by (induct)(auto)
+
+lemma L_flat_Prf2:
+  assumes "s \<in> lang r" shows "\<exists>v. \<turnstile> v : r \<and> flat v = s"
+using assms
+apply(induct r arbitrary: s)
+apply(auto intro: Prf.intros)
+using Prf.intros(2) flat.simps(3) apply blast
+using Prf.intros(3) flat.simps(4) apply blast
+apply (metis Prf.intros(1) concE flat.simps(5))
+apply(subgoal_tac "\<exists>vs::('a val) list. concat (map flat vs) = s \<and> (\<forall>v \<in> set vs. \<turnstile> v : r)")
+apply(auto)[1]
+apply(rule_tac x="Stars vs" in exI)
+apply(simp)
+apply (simp add: Prf_Stars)
+apply(drule Star_string)
+apply(auto)
+apply(rule Star_val)
+apply(auto)
+done
+
+lemma L_flat_Prf:
+  "lang r = {flat v | v. \<turnstile> v : r}"
+using L_flat_Prf1 L_flat_Prf2 by blast
+
+
+section {* Sulzmann and Lu functions *}
+
+fun 
+  mkeps :: "'a rexp \<Rightarrow> 'a val"
+where
+  "mkeps(One) = Void"
+| "mkeps(Times r1 r2) = Seq (mkeps r1) (mkeps r2)"
+| "mkeps(Plus r1 r2) = (if nullable(r1) then Left (mkeps r1) else Right (mkeps r2))"
+| "mkeps(Star r) = Stars []"
+
+fun injval :: "'a rexp \<Rightarrow> 'a \<Rightarrow> 'a val \<Rightarrow> 'a val"
+where
+  "injval (Atom d) c Void = Atm d"
+| "injval (Plus r1 r2) c (Left v1) = Left(injval r1 c v1)"
+| "injval (Plus r1 r2) c (Right v2) = Right(injval r2 c v2)"
+| "injval (Times r1 r2) c (Seq v1 v2) = Seq (injval r1 c v1) v2"
+| "injval (Times r1 r2) c (Left (Seq v1 v2)) = Seq (injval r1 c v1) v2"
+| "injval (Times r1 r2) c (Right v2) = Seq (mkeps r1) (injval r2 c v2)"
+| "injval (Star r) c (Seq v (Stars vs)) = Stars ((injval r c v) # vs)" 
+
+
+section {* Mkeps, injval *}
+
+lemma mkeps_nullable:
+  assumes "nullable r" 
+  shows "\<turnstile> mkeps r : r"
+using assms
+by (induct r) 
+   (auto intro: Prf.intros)
+
+lemma mkeps_flat:
+  assumes "nullable r" 
+  shows "flat (mkeps r) = []"
+using assms
+by (induct r) (auto)
+
+
+lemma Prf_injval:
+  assumes "\<turnstile> v : deriv c r" 
+  shows "\<turnstile> (injval r c v) : r"
+using assms
+apply(induct r arbitrary: c v rule: rexp.induct)
+apply(auto intro!: Prf.intros mkeps_nullable elim!: Prf_elims split: if_splits)
+(* Star *)
+apply(rotate_tac 2)
+apply(erule Prf.cases)
+apply(simp_all)[7]
+apply(auto)
+apply (metis Prf.intros(6) Prf.intros(7))
+by (metis Prf.intros(7))
+
+lemma Prf_injval_flat:
+  assumes "\<turnstile> v : deriv c r" 
+  shows "flat (injval r c v) = c # (flat v)"
+using assms
+apply(induct r arbitrary: v c)
+apply(auto elim!: Prf_elims split: if_splits)
+apply(metis mkeps_flat)
+apply(rotate_tac 2)
+apply(erule Prf.cases)
+apply(simp_all)[7]
+done
+
+(* HERE *)
+
+section {* Our Alternative Posix definition *}
+
+inductive 
+  Posix :: "'a list \<Rightarrow> 'a rexp \<Rightarrow> 'a val \<Rightarrow> bool" ("_ \<in> _ \<rightarrow> _" [100, 100, 100] 100)
+where
+  Posix_One: "[] \<in> One \<rightarrow> Void"
+| Posix_Atom: "[c] \<in> (Atom c) \<rightarrow> (Atm c)"
+| Posix_Plus1: "s \<in> r1 \<rightarrow> v \<Longrightarrow> s \<in> (Plus r1 r2) \<rightarrow> (Left v)"
+| Posix_Plus2: "\<lbrakk>s \<in> r2 \<rightarrow> v; s \<notin> lang r1\<rbrakk> \<Longrightarrow> s \<in> (Plus r1 r2) \<rightarrow> (Right v)"
+| Posix_Times: "\<lbrakk>s1 \<in> r1 \<rightarrow> v1; s2 \<in> r2 \<rightarrow> v2;
+    \<not>(\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> (s1 @ s\<^sub>3) \<in> lang r1 \<and> s\<^sub>4 \<in> lang r2)\<rbrakk> \<Longrightarrow> 
+    (s1 @ s2) \<in> (Times r1 r2) \<rightarrow> (Seq v1 v2)"
+| Posix_Star1: "\<lbrakk>s1 \<in> r \<rightarrow> v; s2 \<in> Star r \<rightarrow> Stars vs; flat v \<noteq> [];
+    \<not>(\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> (s1 @ s\<^sub>3) \<in> lang r \<and> s\<^sub>4 \<in> lang (Star r))\<rbrakk>
+    \<Longrightarrow> (s1 @ s2) \<in> Star r \<rightarrow> Stars (v # vs)"
+| Posix_Star2: "[] \<in> Star r \<rightarrow> Stars []"
+
+inductive_cases Posix_elims:
+  "s \<in> Zero \<rightarrow> v"
+  "s \<in> One \<rightarrow> v"
+  "s \<in> Atom c \<rightarrow> v"
+  "s \<in> Plus r1 r2 \<rightarrow> v"
+  "s \<in> Times r1 r2 \<rightarrow> v"
+  "s \<in> Star r \<rightarrow> v"
+
+lemma Posix1:
+  assumes "s \<in> r \<rightarrow> v"
+  shows "s \<in> lang r" "flat v = s"
+using assms
+by (induct s r v rule: Posix.induct) (auto)
+
+
+lemma Posix1a:
+  assumes "s \<in> r \<rightarrow> v"
+  shows "\<turnstile> v : r"
+using assms
+by (induct s r v rule: Posix.induct)(auto intro: Prf.intros)
+
+
+lemma Posix_mkeps:
+  assumes "nullable r"
+  shows "[] \<in> r \<rightarrow> mkeps r"
+using assms
+apply(induct r)
+apply(auto intro: Posix.intros simp add: nullable_iff)
+apply(subst append.simps(1)[symmetric])
+apply(rule Posix.intros)
+apply(auto)
+done
+
+
+lemma Posix_determ:
+  assumes "s \<in> r \<rightarrow> v1" "s \<in> r \<rightarrow> v2"
+  shows "v1 = v2"
+using assms
+proof (induct s r v1 arbitrary: v2 rule: Posix.induct)
+  case (Posix_One v2)
+  have "[] \<in> One \<rightarrow> v2" by fact
+  then show "Void = v2" by cases auto
+next 
+  case (Posix_Atom c v2)
+  have "[c] \<in> Atom c \<rightarrow> v2" by fact
+  then show "Atm c = v2" by cases auto
+next 
+  case (Posix_Plus1 s r1 v r2 v2)
+  have "s \<in> Plus r1 r2 \<rightarrow> v2" by fact
+  moreover
+  have "s \<in> r1 \<rightarrow> v" by fact
+  then have "s \<in> lang r1" by (simp add: Posix1)
+  ultimately obtain v' where eq: "v2 = Left v'" "s \<in> r1 \<rightarrow> v'" by cases auto 
+  moreover
+  have IH: "\<And>v2. s \<in> r1 \<rightarrow> v2 \<Longrightarrow> v = v2" by fact
+  ultimately have "v = v'" by simp
+  then show "Left v = v2" using eq by simp
+next 
+  case (Posix_Plus2 s r2 v r1 v2)
+  have "s \<in> Plus r1 r2 \<rightarrow> v2" by fact
+  moreover
+  have "s \<notin> lang r1" by fact
+  ultimately obtain v' where eq: "v2 = Right v'" "s \<in> r2 \<rightarrow> v'" 
+    by cases (auto simp add: Posix1) 
+  moreover
+  have IH: "\<And>v2. s \<in> r2 \<rightarrow> v2 \<Longrightarrow> v = v2" by fact
+  ultimately have "v = v'" by simp
+  then show "Right v = v2" using eq by simp
+next
+  case (Posix_Times s1 r1 v1 s2 r2 v2 v')
+  have "(s1 @ s2) \<in> Times r1 r2 \<rightarrow> v'" 
+       "s1 \<in> r1 \<rightarrow> v1" "s2 \<in> r2 \<rightarrow> v2"
+       "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> lang r1 \<and> s\<^sub>4 \<in> lang r2)" by fact+
+  then obtain v1' v2' where "v' = Seq v1' v2'" "s1 \<in> r1 \<rightarrow> v1'" "s2 \<in> r2 \<rightarrow> v2'"
+  apply(cases) apply (auto simp add: append_eq_append_conv2)
+  using Posix1(1) by fastforce+
+  moreover
+  have IHs: "\<And>v1'. s1 \<in> r1 \<rightarrow> v1' \<Longrightarrow> v1 = v1'"
+            "\<And>v2'. s2 \<in> r2 \<rightarrow> v2' \<Longrightarrow> v2 = v2'" by fact+
+  ultimately show "Seq v1 v2 = v'" by simp
+next
+  case (Posix_Star1 s1 r v s2 vs v2)
+  have "(s1 @ s2) \<in> Star r \<rightarrow> v2" 
+       "s1 \<in> r \<rightarrow> v" "s2 \<in> Star r \<rightarrow> Stars vs" "flat v \<noteq> []"
+       "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> lang r \<and> s\<^sub>4 \<in> lang (Star r))" by fact+
+  then obtain v' vs' where "v2 = Stars (v' # vs')" "s1 \<in> r \<rightarrow> v'" "s2 \<in> (Star r) \<rightarrow> (Stars vs')"
+  apply(cases) apply (auto simp add: append_eq_append_conv2)
+  using Posix1(1) apply fastforce
+  apply (metis Posix1(1) Posix_Star1.hyps(6) append_Nil append_Nil2)
+  using Posix1(2) by blast
+  moreover
+  have IHs: "\<And>v2. s1 \<in> r \<rightarrow> v2 \<Longrightarrow> v = v2"
+            "\<And>v2. s2 \<in> Star r \<rightarrow> v2 \<Longrightarrow> Stars vs = v2" by fact+
+  ultimately show "Stars (v # vs) = v2" by auto
+next
+  case (Posix_Star2 r v2)
+  have "[] \<in> Star r \<rightarrow> v2" by fact
+  then show "Stars [] = v2" by cases (auto simp add: Posix1)
+qed
+
+
+lemma Posix_injval:
+  assumes "s \<in> (deriv c r) \<rightarrow> v"
+  shows "(c # s) \<in> r \<rightarrow> (injval r c v)"
+using assms
+proof(induct r arbitrary: s v rule: rexp.induct)
+  case Zero
+  have "s \<in> deriv c Zero \<rightarrow> v" by fact
+  then have "s \<in> Zero \<rightarrow> v" by simp
+  then have "False" by cases
+  then show "(c # s) \<in> Zero \<rightarrow> (injval Zero c v)" by simp
+next
+  case One
+  have "s \<in> deriv c One \<rightarrow> v" by fact
+  then have "s \<in> Zero \<rightarrow> v" by simp
+  then have "False" by cases
+  then show "(c # s) \<in> One \<rightarrow> (injval One c v)" by simp
+next 
+  case (Atom d)
+  consider (eq) "c = d" | (ineq) "c \<noteq> d" by blast
+  then show "(c # s) \<in> (Atom d) \<rightarrow> (injval (Atom d) c v)"
+  proof (cases)
+    case eq
+    have "s \<in> deriv c (Atom d) \<rightarrow> v" by fact
+    then have "s \<in> One \<rightarrow> v" using eq by simp
+    then have eqs: "s = [] \<and> v = Void" by cases simp
+    show "(c # s) \<in> Atom d \<rightarrow> injval (Atom d) c v" using eq eqs 
+    by (auto intro: Posix.intros)
+  next
+    case ineq
+    have "s \<in> deriv c (Atom d) \<rightarrow> v" by fact
+    then have "s \<in> Zero \<rightarrow> v" using ineq by simp
+    then have "False" by cases
+    then show "(c # s) \<in> Atom d \<rightarrow> injval (Atom d) c v" by simp
+  qed
+next
+  case (Plus r1 r2)
+  have IH1: "\<And>s v. s \<in> deriv c r1 \<rightarrow> v \<Longrightarrow> (c # s) \<in> r1 \<rightarrow> injval r1 c v" by fact
+  have IH2: "\<And>s v. s \<in> deriv c r2 \<rightarrow> v \<Longrightarrow> (c # s) \<in> r2 \<rightarrow> injval r2 c v" by fact
+  have "s \<in> deriv c (Plus r1 r2) \<rightarrow> v" by fact
+  then have "s \<in> Plus (deriv c r1) (deriv c r2) \<rightarrow> v" by simp
+  then consider (left) v' where "v = Left v'" "s \<in> deriv c r1 \<rightarrow> v'" 
+              | (right) v' where "v = Right v'" "s \<notin> lang (deriv c r1)" "s \<in> deriv c r2 \<rightarrow> v'" 
+              by cases auto
+  then show "(c # s) \<in> Plus r1 r2 \<rightarrow> injval (Plus r1 r2) c v"
+  proof (cases)
+    case left
+    have "s \<in> deriv c r1 \<rightarrow> v'" by fact
+    then have "(c # s) \<in> r1 \<rightarrow> injval r1 c v'" using IH1 by simp
+    then have "(c # s) \<in> Plus r1 r2 \<rightarrow> injval (Plus r1 r2) c (Left v')" by (auto intro: Posix.intros)
+    then show "(c # s) \<in> Plus r1 r2 \<rightarrow> injval (Plus r1 r2) c v" using left by simp
+  next 
+    case right
+    have "s \<notin> lang (deriv c r1)" by fact
+    then have "c # s \<notin> lang r1" by (simp add: lang_deriv Deriv_def)
+    moreover 
+    have "s \<in> deriv c r2 \<rightarrow> v'" by fact
+    then have "(c # s) \<in> r2 \<rightarrow> injval r2 c v'" using IH2 by simp
+    ultimately have "(c # s) \<in> Plus r1 r2 \<rightarrow> injval (Plus r1 r2) c (Right v')" 
+      by (auto intro: Posix.intros)
+    then show "(c # s) \<in> Plus r1 r2 \<rightarrow> injval (Plus r1 r2) c v" using right by simp
+  qed
+next
+  case (Times r1 r2)
+  have IH1: "\<And>s v. s \<in> deriv c r1 \<rightarrow> v \<Longrightarrow> (c # s) \<in> r1 \<rightarrow> injval r1 c v" by fact
+  have IH2: "\<And>s v. s \<in> deriv c r2 \<rightarrow> v \<Longrightarrow> (c # s) \<in> r2 \<rightarrow> injval r2 c v" by fact
+  have "s \<in> deriv c (Times r1 r2) \<rightarrow> v" by fact
+  then consider 
+        (left_nullable) v1 v2 s1 s2 where 
+        "v = Left (Seq v1 v2)"  "s = s1 @ s2" 
+        "s1 \<in> deriv c r1 \<rightarrow> v1" "s2 \<in> r2 \<rightarrow> v2" "nullable r1" 
+        "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> lang (deriv c r1) \<and> s\<^sub>4 \<in> lang r2)"
+      | (right_nullable) v1 s1 s2 where 
+        "v = Right v1" "s = s1 @ s2"  
+        "s \<in> deriv c r2 \<rightarrow> v1" "nullable r1" "s1 @ s2 \<notin> lang (Times (deriv c r1) r2)"
+      | (not_nullable) v1 v2 s1 s2 where
+        "v = Seq v1 v2" "s = s1 @ s2" 
+        "s1 \<in> deriv c r1 \<rightarrow> v1" "s2 \<in> r2 \<rightarrow> v2" "\<not>nullable r1" 
+        "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> lang (deriv c r1) \<and> s\<^sub>4 \<in> lang r2)"
+        by (force split: if_splits elim!: Posix_elims simp add: lang_deriv Deriv_def)   
+  then show "(c # s) \<in> Times r1 r2 \<rightarrow> injval (Times r1 r2) c v" 
+    proof (cases)
+      case left_nullable
+      have "s1 \<in> deriv c r1 \<rightarrow> v1" by fact
+      then have "(c # s1) \<in> r1 \<rightarrow> injval r1 c v1" using IH1 by simp
+      moreover
+      have "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> lang (deriv c r1) \<and> s\<^sub>4 \<in> lang r2)" by fact
+      then have "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> (c # s1) @ s\<^sub>3 \<in> lang r1 \<and> s\<^sub>4 \<in> lang r2)" 
+         by (simp add: lang_deriv Deriv_def)
+      ultimately have "((c # s1) @ s2) \<in> Times r1 r2 \<rightarrow> Seq (injval r1 c v1) v2" using left_nullable by (rule_tac Posix.intros)
+      then show "(c # s) \<in> Times r1 r2 \<rightarrow> injval (Times r1 r2) c v" using left_nullable by simp
+    next
+      case right_nullable
+      have "nullable r1" by fact
+      then have "[] \<in> r1 \<rightarrow> (mkeps r1)" by (rule Posix_mkeps)
+      moreover
+      have "s \<in> deriv c r2 \<rightarrow> v1" by fact
+      then have "(c # s) \<in> r2 \<rightarrow> (injval r2 c v1)" using IH2 by simp
+      moreover
+      have "s1 @ s2 \<notin> lang (Times (deriv c r1) r2)" by fact
+      then have "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = c # s \<and> [] @ s\<^sub>3 \<in> lang r1 \<and> s\<^sub>4 \<in> lang r2)" 
+        using right_nullable 
+        apply (auto simp add: lang_deriv Deriv_def append_eq_Cons_conv)
+        by (metis concI mem_Collect_eq)
+      ultimately have "([] @ (c # s)) \<in> Times r1 r2 \<rightarrow> Seq (mkeps r1) (injval r2 c v1)"
+      by(rule Posix.intros)
+      then show "(c # s) \<in> Times r1 r2 \<rightarrow> injval (Times r1 r2) c v" using right_nullable by simp
+    next
+      case not_nullable
+      have "s1 \<in> deriv c r1 \<rightarrow> v1" by fact
+      then have "(c # s1) \<in> r1 \<rightarrow> injval r1 c v1" using IH1 by simp
+      moreover
+      have "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> lang (deriv c r1) \<and> s\<^sub>4 \<in> lang r2)" by fact
+      then have "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> (c # s1) @ s\<^sub>3 \<in> lang r1 \<and> s\<^sub>4 \<in> lang r2)" by (simp add: lang_deriv Deriv_def)
+      ultimately have "((c # s1) @ s2) \<in> Times r1 r2 \<rightarrow> Seq (injval r1 c v1) v2" using not_nullable 
+        by (rule_tac Posix.intros) (simp_all) 
+      then show "(c # s) \<in> Times r1 r2 \<rightarrow> injval (Times r1 r2) c v" using not_nullable by simp
+    qed
+next
+  case (Star r)
+  have IH: "\<And>s v. s \<in> deriv c r \<rightarrow> v \<Longrightarrow> (c # s) \<in> r \<rightarrow> injval r c v" by fact
+  have "s \<in> deriv c (Star r) \<rightarrow> v" by fact
+  then consider
+      (cons) v1 vs s1 s2 where 
+        "v = Seq v1 (Stars vs)" "s = s1 @ s2" 
+        "s1 \<in> deriv c r \<rightarrow> v1" "s2 \<in> (Star r) \<rightarrow> (Stars vs)"
+        "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> lang (deriv c r) \<and> s\<^sub>4 \<in> lang (Star r))" 
+        apply(auto elim!: Posix_elims(1-5) simp add: lang_deriv Deriv_def intro: Posix.intros)
+        apply(rotate_tac 3)
+        apply(erule_tac Posix_elims(6))
+        apply (simp add: Posix.intros(6))
+        using Posix.intros(7) by blast
+    then show "(c # s) \<in> Star r \<rightarrow> injval (Star r) c v" 
+    proof (cases)
+      case cons
+          have "s1 \<in> deriv c r \<rightarrow> v1" by fact
+          then have "(c # s1) \<in> r \<rightarrow> injval r c v1" using IH by simp
+        moreover
+          have "s2 \<in> Star r \<rightarrow> Stars vs" by fact
+        moreover 
+          have "(c # s1) \<in> r \<rightarrow> injval r c v1" by fact 
+          then have "flat (injval r c v1) = (c # s1)" by (rule Posix1)
+          then have "flat (injval r c v1) \<noteq> []" by simp
+        moreover 
+          have "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> lang (deriv c r) \<and> s\<^sub>4 \<in> lang (Star r))" by fact
+          then have "\<not> (\<exists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> (c # s1) @ s\<^sub>3 \<in> lang r \<and> s\<^sub>4 \<in> lang (Star r))" 
+            by (simp add: lang_deriv Deriv_def)
+        ultimately 
+        have "((c # s1) @ s2) \<in> Star r \<rightarrow> Stars (injval r c v1 # vs)" by (rule Posix.intros)
+        then show "(c # s) \<in> Star r \<rightarrow> injval (Star r) c v" using cons by(simp)
+    qed
+qed
+
+
+section {* The Lexer by Sulzmann and Lu  *}
+
+fun 
+  lexer :: "'a rexp \<Rightarrow> 'a list \<Rightarrow> ('a val) option"
+where
+  "lexer r [] = (if nullable r then Some(mkeps r) else None)"
+| "lexer r (c#s) = (case (lexer (deriv c r) s) of  
+                    None \<Rightarrow> None
+                  | Some(v) \<Rightarrow> Some(injval r c v))"
+
+
+lemma lexer_correct_None:
+  shows "s \<notin> lang r \<longleftrightarrow> lexer r s = None"
+using assms
+apply(induct s arbitrary: r)
+apply(simp add: nullable_iff)
+apply(drule_tac x="deriv a r" in meta_spec)
+apply(auto simp add: lang_deriv Deriv_def)
+done
+
+lemma lexer_correct_Some:
+  shows "s \<in> lang r \<longleftrightarrow> (\<exists>v. lexer r s = Some(v) \<and> s \<in> r \<rightarrow> v)"
+using assms
+apply(induct s arbitrary: r)
+apply(auto simp add: Posix_mkeps nullable_iff)[1]
+apply(drule_tac x="deriv a r" in meta_spec)
+apply(simp add: lang_deriv Deriv_def)
+apply(rule iffI)
+apply(auto intro: Posix_injval simp add: Posix1(1))
+done 
+
+lemma lexer_correctness:
+  shows "(lexer r s = Some v) \<longleftrightarrow> s \<in> r \<rightarrow> v"
+  and   "(lexer r s = None) \<longleftrightarrow> \<not>(\<exists>v. s \<in> r \<rightarrow> v)"
+apply(auto)
+using lexer_correct_None lexer_correct_Some apply fastforce
+using Posix1(1) Posix_determ lexer_correct_Some apply blast
+using Posix1(1) lexer_correct_None apply blast
+using lexer_correct_None lexer_correct_Some by blast
+
+
+end
\ No newline at end of file